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Question
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
Solution
No.
* is not a binary operation on Z.
Reason: Since m, n ∈ Z.
So, m, n can be negative also.
Now, if n is negative (Le.) say n = – k where k is +ve.
Then mn = m–k = `1/"m"^"k"` ∈ Z.
Similarly, when m is negative then nm ∉ Z.
∴ m * n ∉ Z.
⇒ * is not a binary operation on Z.
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